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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Trig Identities from Properties of Complex CalculationsFormulas for cos(2A) and sin (2A)From cis(2A) = cis(A)*cis(A) we found
Therefore comparison of rectangular coordinates/components (or real and imaginary parts) yields the double angle formulas
Formulas for cos(nA) and sin (nA), the case n =3.Observe
Therefore
Equality of corresponding real and imaginary parts gives
Exponential or cis Expressions for Trig FunctionsFrom the two equations exp(iA) = [cos(A),sin(A)] = cos(A) + i sin(A) exp(-iA) = [cos(-A),sin(-A)] = cos(A) - i sin(A) we see exp(iA) + exp(-iA) = 2 cos(A) and exp(iA) - exp(-iA) = 2i sin(A) Therefore (1/2)[exp(iA) + exp(-iA)] = cos(A) and (1/2i) [exp(iA) - exp(-iA)] = sin(A) Therefore tan(A) = sin(A)/cos(A) = (1/i) { [exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)] } or tan(A) = sin(A)/cos(A) = -i{[exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)]} So all trig functions may be expressed in terms of exp(iA) and exp(-iA). The substitution of exp(iA) and exp(-iA) expressions for trig functions turns the proof or derivation of trig identities into simple algebraic exercises involving complex numbers and the add the angles rule for multiplication of exp(iA) with exp(iB). While highschool students may be taken through the exercises of proving trig identities before meeting complex numbers, the above quick explanation of complex numbers and its links to trig imply a quick route through highschool mathematics courses on algebra and trig. In retrospect, the presentation of trig in highschool has been harder than need-be. More motivation and more applications of trig, tied to right triangles and complex numbers, come from engineering and physics. The description of electric currents and devices depends on sine and cosines, or more simply, if you know complex numbers and exp(iA) = exp(iA). The latter functions also appear in the theoretical treatment of statistics and of higher mathematics (analysis). . Exponentials of Complex NumbersIf you have met the exponential ea = exp (a) of a real number a then you may be please to know that the exponential of a complex number a + ib (where a and b are real) is given by
Take this as a computational definition. Higher mathematics in science, engineering, statistics and economics is often best seen with a knowledge of the exponentials ea or eib or ea+ib . One may write ex instead of exp(x), or vice versa.
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